Two small objects, P of mass \(m \mathrm {~kg}\) and Q of mass \(k m \mathrm {~kg}\), slide on a smooth horizontal plane. Initially, P and Q are moving in the same straight line towards one another, each with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
After a direct collision with P , the direction of motion of Q is reversed and it now has a speed of \(\frac { 1 } { 3 } u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of P is now \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where the positive direction is the original direction of motion of P .
Draw a diagram showing the velocities of P and Q before and after the impact.
By considering the linear momentum of the objects before and after the collision, show that \(v = \left( 1 - \frac { 4 } { 3 } k \right) u\).
Hence find the condition on \(k\) for the direction of motion of P to be reversed.
The coefficient of restitution in the collision is 0.5 .
Show that \(v = - \frac { 2 } { 3 } u\) and calculate the value of \(k\).
Particle \(A\) has a mass of 5 kg and velocity \(\binom { 3 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Particle \(B\) has mass 3 kg and is initially at rest. A force \(\binom { 1 } { - 2 } \mathrm {~N}\) acts for 9 seconds on B and subsequently (in the absence of the force), \(A\) and \(B\) collide and stick together to form an object \(C\) that moves off with a velocity \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\).
Show that \(\mathbf { V } = \binom { 3 } { - 1 }\).
The object C now collides with a smooth barrier which lies in the direction \(\binom { 0 } { 1 }\). The coefficient of restitution in the collision is 0.5 .